MECANICA DE FLUIDOS 1. PROPIEDAD PROPIEDADES ES Y COMPORTAMI COMPORTAMIENTO ENTO DE LOS FLUIDOS FLUIDOS Densidad absoluta paa l!"uidos *+&%e ρ = *e&s'%a% m = Masa %el fl'%+
m kg . lb , ρ = 3 v m3 ft f t
KG
ρ
H 2O
= 1 00
Atm.
ρ
=
H 2O
= 101.937
3
m
!2.4
-+lme& %el fl'%+
lb
=
3
ft
1.94
UTM 3
m
=1
gr 3
cm
Para Para el H2O a 4.4ºC y 1
s/g
ft
Para el H 2 O a 39º"
3
y 14.7
lb 2
pg pg
E#UI$ALENCIAS 1 sl slug
1
=14 .#9 kg =32 .1!# lb
kg
m
= 0.101937 3
UTM UT M
m
3
2
1UTM UTM
=
1
1UTM
=1
kg − seg seg m
Densidad absoluta paa %ases Pv
MRT M RT
=
p = ρ =
m v P
RT ) P = ρ (T
(T
ρ aire
Tem$eratra %e refere&c'a 20ºC
*+&%e ( - C+&sta&te %el gas
= 1.20#9
kg
m
3
= 0.07#3
l
ft
3
Para el a're a 20ºC y 1.033
kg
cm
2
ρ aire = 0.01
lb
=
3
ft
1.2977
g
m
Para el a're a 32º" y 14,7
3
lb 2
pg
$olu&en Espe'!(i'o v V = m
1
=
3
m , ft ) m 3
g
ρ
lb
3
UTM
V =+lme& es$ecf'c+
*+&%e
- +lme& t+tal
3
3
3
1 m = 1 X −3 m = 9.09 X −3 m = = 0 . 001 V H 2O 10 Kg 10 UTM Kg Kg 1000
m
3 3
1
= 0.014#9 m
=
slg
V
=
a ir e
3
= 0. 2 9m4 = .1 3 ! m3
1
k g 1.2 0 # 73
k g
3
1 = 1 2.1 0 m
U T M
lb
3
3
℘=
v
=
mg V
ω = m. g
= ρ . g
s l + g
g N lb N kg 2 3 , 3 , 3 , 3 , 2 m m ft ft m seg *+&%e
ω = Pes+
m = Masa g = rae%a%
E"ui)alen'ias m m 1 kg = Kgx9.1 = 9.1 Kg 2 f seg seg
=
9.1 N
f t
= 1 3.2
m
ω
ft 0.01!02
lb
3
ft seg
1lb f = 1lbx 32.2
2
=
ft
32.2lb
2
seg
$ISCOSIDAD A*SOLUTA + $ISCOSIDAD DIN,MICA µ
*+&%e A - A're %e la $laca e& m+'m'e&t+ - el+c'%a% %e $laca 8- es$es+r %e la $laca %e fl'%+s µ = "act+r %e $r+$+rc'+&al'%a% + c+ef'c'e&te %e 'sc+s'%a%
=τ v
y
τ =
A
A
= µ
v
v y
y
τ µ
(a$'%e %e %ef+rmac'& a&glar + gra%'e&te %e el+c'%a%
= :sfer+ c+rta&te %el fl'%+
P+'ses . . N seg kg seg . lb.seg ) , , m m ft 'sc+s'%a% c'&em6t'ca
5e m'%e e&
f
2
V=
3
(elac'& e&tre 'sc+s'%a% 2 2 2 ft ρ !m )m ) V= seg seg seg
2 2 cm , ft seg seg
µ ρ
A 20 ºC =
µ H 2O
2
µ ρ
2 3 Cm , ft seg seg 1ar.
y
!
−
=
!
1.002 x10 P" seg .
VH2O- 1.01 x
10
−!
!m
2
−
2
µ aire = 1.19 x10 P".seg
ft
) seg seg
2
Vaire- 1# .1 x
10
−!
m
seg
E"ui)alen'ias. N seg . µ = 1 2
1 p"ise
=1
m
- 1 p" .seg
0.1
=
gr cm .seg
kg . m seg .
kg
- 1 m. seg 1 p" = 1
0.1 P" . seg
=
N
m
2
=
N seg . 0 .1 2
m
=
N seg . 1 P" seg . =1 2
1 Ce&t'$+'se - 1cP - 0.01P - 0.001 $+.seg
m
Pesi-n ; ) Kg ) KPa ) bar ) lb 2 2 m cm 2 pg
P =
A
Pesi-n at&os(i'a .
- 1 Atmsfera - 7!0 t +rr - 7!0 mm Hg - 1.013 ar - 1.033 g/cm 2 -10.33 m. c. a Pa - 14.7 l/Pg2 - 29.92 Pg Hg - 33091 $. c. a. <$'es c+lm&a %e aga=
E"ui)alen'ias
1 Pa = 1
N
m
1bar = 1.02
1
Kg
cm
2
2
= 1.02 x10
Kg
−#
Kg
cm
2
#
= 10 Pa = 100 KPa 2
cm
=14 .22
lb 2
Pg
=10 m.c.a
Tensi-n supe(i'ial N kG . ) T # m m
*+&%e te&s'& s$erf'c'al " - "era + :&erga s$erf'c'al ? - l+&g't%
T =
Paa una &ol'ula es(i'a *+&%e Pr P - Permetr+ = 2 r - (a%'+ T
TH2O -
0.074
N
=
m
7.#43 x10
3
−
Kg m
a 20º C
Te&peatua # º ! = ( º − 32 )
Te&peatua absoluta º K =º ! + 273
9
9
º =
#
º R =º + 4!0
º ! + 32
/. EST,TICA DE FLUIDOS ω )℘= ν A
ρ =
*+&%e - +lme& - A.@
= P . A) ω =℘v
P A =℘ν
∴P =
℘ν
⇒ P =℘.$Pres'& A P = ρ g .$ @'%r+st6t'ca
$ + P atm
P1- ρ g
& i
ρ =
m
Pres'& e& cal'er $&t+
1
∴
m = ρ ν = ρ A%y
ν
Paa el e"uilibio en la die''i-n 0Y
P A − ( ρ + % P ) A y− ρ g A y% y = 0 y
−
Resol)iendo2
−
Inte%ando2
P P 1
−
2
A%upando2
(
g y
=
ρ
%P
= g%y
1 = g y2 %y P 2 y1 ρ %P P 1
∫
∫
y )
− 2
1
:cac'& $ara fl'%+s l'%+s '&c+m$re&s'les
P + g y = P + g y 1
ρ
2
1
ρ
2
*+&%e P - Pres'& ρ = *e&s'%a% g - rae%a% y - Altra
Par a & s+l+ esta%+ P ρ
P ρ
+ gy = !
P ρ g
+ y =
P + y ρ g
:cac'& f&%ame&tal %e la :st6t'ca %e fl'%+s, 1º f+rma
+ gy = !
*''%'e&%+ e&tre g
!
C
5'
g
g
=$
P
+ y = $2º f+rma %e la ecac'& ℘
= $)
*+&%e @- Altra P'e+mBtr'ca P g ρ
+ y = $)
g Mlt'$l'ca&%+ $+r ρ
3º f+rma %e la ecac'& :&erga %e $res'&
P + ρ gy = ρ g$ P +℘ y =℘$
Pin'ipio de Pas'al f *+&%e P = a = P = A f - "era a$l'ca%a e& el Bm+l+ me&+r 1
2
" - "era +te&'%a e& el em+l+ may+r
f = P 1 a = f = P 2 A
P P =
1
f
=
a
f π
2
2
" A
=
%
'
π
2
4
4 4 f π
*+&%e P1 - Pres'& eerc'%a e& el em+l+ me&+r P2 - Pres'& +te&'%a e& el em+l+ may+r a - Drea %el em+l+ me&+r A - Drea %el em+l+ may+r % - *'6metr+ %el em+l+ me&+r * - *'6metr+ %el em+l+ may+r
2
4
=
%
'
π
2
Pesi-n 3idost4ti'a sobe una supe(i'ie plana su&e%ida
&e( ∝ =
$ y
∴ @ = y se& ∝
De la pesi-n
Di(een'iando la (ue5a
P A = P . A
% = P ( %a γ $.%A ρ g$ .%a
=
=
= P ( . A Sustitu6endo
E&tegra&%+
% = P ( .%A = γ .$%
∫ % =
γ &e(
∝ ∫ )%A
% = γ ( & ∝ ) % R R
=γ se( α ∫ y%A =γ se( α <) ce A
Paa el 'entoide 0Ce se( α =
$ce ) ce
∴
@ ce
=
) ce se( α
5st'tye&%+) R = γ se( α ( ) ce A ) = γ ( ) ce se( α ) A R
$ A
= γ ce
%"(%e R
uer*a res ul ta& te
=
*e la ecac'& %e M+me&t+ ,T-. "era F ra+ %e $ala&ca - " F %T - %" F 8 T
=[γ ()se( α )%A ]) =γ ) 2 se( α %A α %T =γ se( α ) 2 %A %T
/(tegra(%" se( α ∫ ) %A ∫%T =γ
&i
T =γ se( γ /
= M"me(t" %e
2
%"(%e /
∫ ) %A = / 2
i(ercia
:l M+me&t+ e& el ce&tr+ %e $res'& GC$ Tc$ - "( F 8c$
)
Egala&%+ M+me&t+s
Tc$ - E
?a '&tegral ∫ )%A =)ceA
∴ R) cp = γ /se( α ) cp
=
/ ) ce A
= γ se( α () ce A) =
si R
∴ ) cp −) ce =
/ ce ) ce A
/ ) ce A
P"r el te"rema %e tra(sfere( cia
Pesi-n 7idost4ti'a sobe una supe(i'ie 'u)a :& el ce&tr+ %e grae%a%, la fera @+r'+&tal " H es "H - "ce - PceA
P e r ") c e = ( ) + ∴ H = γ ) c A e
# 2
)
H = γ ( ) + #2 ) A = γ ( ) + #2 ) s 0 % " ( %#2e= % i st a &c i a a l tcr e" ( % e s la$e r f i c i er vc *e la ecac'& ) cp −) ce = 0s 3 / ce = 12
/ ce ) ce A
, el 6rea %e la secc'& cra ,
s 2 ∴) cp −) ce = 12 ) ce
Calcl+ e la fera ert'cal " , s' " - I y
γ =
R
0 v
=
) 0 = γ v
+ v H 2
2
A0 ∴ v = γ y
ta& 2 = v H
Paa una ota'i-n de (luidos8 la pesi-n es ) P = P 0 − ρ g* + 12 ρ r 2 0 2 %"(%e , 0 = aceleraci"( a(gular r = ra%i" e( el e1e %e r "taci"( P 0 − P r 2 0 2 + * = ρ g 2 g
*'str'c'& %e $res'+&es res$ect+ a e& f+rma l'&eal, y $aral'ca res$ect+ a Gr
E'ua'iones de balan'e en (o&a inte%al 6 di(een'ial C+&serac'& %e la masa %m = 0 = % ρ %v + sc ρ v%A %+&%e - el+c'%a% %el fl'%+ vc ∫ %T ∫ %t sist sc - s$erf'c'e %e c+&tr+l c - +lme& % c+&tr+l ρ ∴∫ vc ∂ %v + ∑( ρ Av ) sal − ∑( ρ Av ) e(t = 0 :cac'& $ara & c c+& e&tra%as y ∂t sal'%as &'%'me&s'+&ales
∑( Av ) = ∑( Av ) para flu1" estaci"(a ∑m = ∑m c"(servac i"( %e la masa ρ
ρ
e(t
0
sal
ri"
0
e(t
sal
0
∫ ( 6v )%A lu1" %e masa si e(t y s al (" s"( ∑( vA ) = ∑(vA ) Para flu1" i(c"mpres ible 4 = vA = ∑4 = ∑4 lu1" v"lu m5tric" m=
sal
e(t
sal
%"(%e 4
vm = ρ m
u(i %'m e(si"(ales
e(t
= vA 3cuaci"( %
4 1 = A v%A A
∫
e c"(ti(ui %a%
%"(%e vm = vel"ci%a%me%ia
= A1 ∫ ρ %A &i la %e(i%a% ar ia c"( la sec ci7(
E'ua'iones de la Cantidad de &o)i&iento po la Se%unda Le6 de Ne9ton % ( mv ) = ∑ = % ∫ v ρ %v + ∫ v ρ ( vr ( )%A %+&%e <r &= ect+r &'tar'+ %t
%t vc
sc
"era e& las %'recc'+&es
x = − ∂∂ P %x%y%*+ ρ g x %x%y% x y = − ∂∂ P %x%y%*+ ρ g y % x % y% y * = − ∂∂ P %x%y%*+ ρ g * %x%y% * = xi + y1 + *k M =
%H %t
*+&%e <',,= ect+res &'tar'+s
Conse)a'i-n del Mo&ento Lineal
%"(%e9 %H = M"me(t" ci(5tic" = r 8v r = ra%i" v = vel"ci%a% li(eal
% M = / %t 0
/ = M"m e(t" (ercial i 0 = vel"ci%a% a(gular
E'ua'i-n de *enoulli Para %e&s'%a% c+&sta&te
P ρ
2
+ 12 v + g * = !
5' l+s $&t+s %e e&tra%a y sal'%a est6& e& la m'sma l&ea %e c+rr'e&te
P 1 + 12 v12 ρ + ρ g *1 = P 2 + 12 v22 ρ + ρ g *2 e c u a c i " e( :% e r ( "i E'ua'i-n de la ene%!a en (o&a di(een'ial Para las c++r%e&a%as
4 x % y % =* [ 4 x + ∂∂ x 4 x % x] % y %
4 y % x % =* [ 4 y + ∂∂ y 4 y % y] % x % 4 * % x % y= [ 4 * + ∂∂ * 4 * % *] % x % FLU:O EN CONDUCTOS Para fl+ ?am'&ar y Trle&t+ # e % # e %
= 0.0 !( e # am '&a r 1 !
= 4.4 ( e
T u r b l& u t"
%+&%e ?e - ?+&g't% %e e&tra%a (e - ;mer+ %e (ey&+l%s % - *'6metr+ %e c+&%ct+
Pedidas de 'a%a
$ f = ( < 1 − < 2 ) + ( ρ g 1 − ρ g 2 ) = ∇ < + P
P
∇ P ρ g
% " ( % e f ; =$ P 5 r % i% a c%aerga
$ f = ρ 4 g τ #% Pr+$+rc'+&al al esfer+ %e c+rta%ra e& la $are% %el c+&%ct+ Fa'to de (i''i-n $ f
=
f =
f = 1 f
08#
f
# %
τ 0 2 ρ v !4 (e
v2 2 g
%"(%e f > 0
=
=
fact"r %e fricci"(
perimetr" %el c"(%u ct"= val"r me%i"
para= lu1" #a m'& ar
= 2 l+g ( (e f 08# ) − 08
para= lu1" Turbu l& t"
Po Dia%a&a de Mood6 1 = − 2 l+g 3.% 7 + (e2. f #10.# %"(%e;? = altura %e rug"si%a% 0.# f
(
1 f
0.#
)
ε
1.11 !.9 aci7( %e (e y("l%s = − 1. l+g (e + ( 3.7 ) P"r c" rrel ε
%
Di4&eto 7id4uli'o '$ = 4 P A = 4 R$ %+&%e, * @ = %iametr"Hi%r@ulic" P = perim etr"m"1a%" ( @ = ra%i" $i%r@ulic"
ANALISIS DIMENSIONAL Y TEORIA DE SEME:AN;A *e las 4 %'me&s'+&es 6s'ca M - masa, ? - l+&g't%, T - t'em$+,
- tem$eratra
5e estalece & s'stema M?T
) alg&+s al+res e& la s'g'e&te tala
Flues
τ x
= 2 µ ∂∂ xu )τ y = 2 µ ∂∂ yv )τ 9 = 2 µ ∂∂ *0
τ x y = τ y x = µ
( yu + yv ) ∂ ∂
∂ ∂
τ x * = τ * x = µ
( ∂∂ x0 + ∂∂ *u )
τ y * = τ * y = µ
( *v + y0 ) ∂ ∂
∂ ∂
Capa li&ite paa (lu
! f =
0 8! !4 ( e0 8#
%"(%e;! f = c"eficie(t e %e fricc i7( x = %ista &cia e( el 1e x
Capa li&ite paa (lu
=
0.1! 0.14 2
(e
! f =
0.0 2 7 0.14 2
(e
Coe(i'iente de sustenta'i-n ! # = ρ #v A 0.#
! % =
2
' 2 0.# ρ v A